We point out that when \(p=\infty \) , the last two equalities in the above might not be true without certain type of measurability (see Example 5.0.10 in [1]} and Example 1.42 in [2]}). Therefore, we do not mention the above with \(p=\infty \)  to avoid some technicalities. Note that processes in the above spaces are not necessarily \(\mathbb {F}\) -adapted. The subset of, say, \(L^p_{{\cal F}_T}(\Omega ;L^q(0,T;\mathbb {H}))\) , consisting of all \(\mathbb {F}\) -progressively measurable processes, is denoted by \(L^p_\mathbb {F}(\Omega ;\)  \(L^q(0,T;\mathbb {H}))\)  (having subscript \(\mathbb {F}\)  instead of \({\cal F}_T\) ). All the \(\mathbb {F}\) -progressively measurable version of the other spaces listed in the above can be denoted similarly; for example, \(C_\mathbb {F}([0,T];L^p(\Omega ;\mathbb {H}))\) , and so on. Finally, in the above, \([0,T]\)  can be replaced by any \([t,T]\) , and again, when the range space \(\mathbb {H}\)  is clear from the context, we will omit \(\mathbb {H}\) ; for example, \(L^p_\mathbb {F}(\Omega ;C[t,T])\) , and so on.
